1. **Constructing the right triangle $\triangle DEF$:**
- We are given the lengths $AB$ and $BC$ and the fact that the medians to these sides are perpendicular.
- First, we need to find the lengths $DF$ and $DE$ using the given formulas:
$$DF = \sqrt{\frac{BC^2}{15} - \frac{AB^2}{60}}$$
$$DE = \sqrt{\frac{AB^2}{15} - \frac{BC^2}{60}}$$
- Construct a right triangle $\triangle DEF$ with $\angle D = 90^\circ$, $DF$ as one leg, and $DE$ as the other leg.

2. **Placing points $A$ and $C$:**
- Let $A$ be on $DE$ such that $D$ is between $A$ and $E$.
- Let $C$ be on $DF$ such that $D$ is between $C$ and $F$.
- Ensure that $\triangle ADC$ is similar to $\triangle EDF$ and that the ratio $\frac{AC}{FE} = 2$.

3. **Constructing the triangle $\triangle ABC$:**
- Draw a line through $A$ and $F$.
- Draw a line through $C$ and $E$.
- Let the intersection of these two lines be $B$.

4. Verification:
- Verify that the medians to sides $AB$ and $BC$ are perpendicular.
- This can be done by checking that the medians intersect at the centroid and form a right angle.

By following these steps, we have constructed the triangle $\triangle ABC$ with the given properties.

$\blacksquare$